
There exist \(\hat{\theta }_{TW} \in E\)  with \(\partial \ell (\hat{\theta }_{TW}; \mathbf {y}) /\partial \theta = \mathbf {0}\)  such that \(\hat{\theta }_{TW} \overset{p}{\rightarrow } \theta _{L^2}\) , as \(n \rightarrow \infty \) .
This follows from the definition of \(\hat{\theta }_{TW}\)  and, under conditions A1 to A5 and B1 to B4, Theorem 1 of [1]} (which shows \(\hat{\theta }_{TW} \overset{p}{\rightarrow } \theta _{L^2}\) ).

\(\frac{\partial ^2 \ell (\hat{\theta }_{TW}; \mathbf {y})}{\partial \theta \partial \theta ^{ \mathrm {\scriptscriptstyle T} }} \overset{p}{\rightarrow } V.\)
Let \(S_n(\mu ) = \frac{1}{n}\sum _{i=1}^n \left(y_i - \mu (\mathbf {x}_i)\right)^2 + \lambda _n \Vert \mu \Vert ^2_{\mathcal {H}_\kappa (\mathcal {X})}\) . Now \(\hat{\mu }(\cdot )\)  minimises \(S_n\)  over \(\mathcal {H}_\kappa (\mathcal {X})\) . Under C3, and since \(\hat{\theta }_{TW}\)  is consistent for \(\theta _{L^2}\) , it means \(\partial ^2 \eta (\hat{\theta }_{TW}, \cdot )/\partial \theta \partial \theta ^{ \mathrm {\scriptscriptstyle T} }\in \mathcal {H}_\kappa (\mathcal {X})\)  for sufficiently large \(n\) . Let \(\langle \mu _1 , \mu _2 \rangle _{\mathcal {H}_\kappa (\mathcal {X})}\)  denote the inner product of \(\mathcal {H}_\kappa (\mathcal {X})\) , with \(\Vert \mu _1 \Vert _{\mathcal {H}_\kappa (\mathcal {X})}^2 = \langle \mu _1 , \mu _1 \rangle _{\mathcal {H}_\kappa (\mathcal {X})}\) . For \(u,v=1,\dots ,p\) , we have
\(0 & = & \left. \frac{\partial }{\partial t} S_n \left( \hat{\mu }(\cdot ) + t \frac{\partial ^2 \eta (\hat{\theta }_{TW}, \cdot )}{\partial \theta _u \partial \theta _v} \right)\right|_{t=0},\\& = & \frac{2}{n} \sum _{i=1}^n \left(\hat{\mu }(\mathbf {x}_i) - y_i \right) \frac{\partial ^2 \eta (\hat{\theta }_{TW}, \mathbf {x}_i)}{\partial \theta _u \partial \theta _v} + 2 \lambda _n \left\langle \hat{\mu }(\cdot ) , \frac{\partial ^2 \eta (\hat{\theta }_{TW}, \cdot )}{\partial \theta _u \partial \theta _v} \right\rangle _{\mathcal {H}_\kappa (\mathcal {X})},\\& = & \frac{2}{n} \sum _{i=1}^n \left(\hat{\mu }(\mathbf {x}_i) - \mu (\mathbf {x}_i) \right) \frac{\partial ^2 \eta (\hat{\theta }_{TW}, \mathbf {x}_i)}{\partial \theta _u \partial \theta _v} - \frac{2}{n} \sum _{i=1}^n e_i \frac{\partial ^2 \eta (\hat{\theta }_{TW}, \mathbf {x}_i)}{\partial \theta _u \partial \theta _v}\\& & \qquad \qquad + 2 \lambda _n \left\langle \hat{\mu }(\cdot ) , \frac{\partial ^2 \eta (\hat{\theta }_{TW}, \cdot )}{\partial \theta _u \partial \theta _v} \right\rangle _{\mathcal {H}_\kappa (\mathcal {X})},\\& = & 2 \left( C_n + D_n + E_n \right).\) 
Following the same arguments as in the proof of Theorem 1 in [1]}, with \(\partial ^2 \eta (\hat{\theta }_{TW}, \cdot )/\partial \theta _u \partial \theta _v\)  in place of \(\partial \eta (\hat{\theta }_{TW}, \cdot )/\partial \theta _u\) , we obtain
\(C_n & = & \int _{\mathcal {X}} \left(\hat{\mu }(\mathbf {x}) - \mu (\mathbf {x}) \right) \frac{\partial ^2 \eta (\hat{\theta }_{TW}, \mathbf {x})}{\partial \theta _u \partial \theta _v} \mathrm {d}\mathbf {x}+ o_p (n^{-1/2}),\\D_n & = & \frac{1}{n}\sum _{i=1}^n e_i \frac{\partial ^2 \eta (\theta _{L^2}, \mathbf {x}_i)}{\partial \theta _u \partial \theta _v} + o_p (n^{-1/2}),\\& = & o_p (1), \qquad \qquad \mbox{by the weak law of large numbers,}\\E_n & \le & o_p(n^{-1/2}).\) 
Therefore
\(\int _{\mathcal {X}} \hat{\mu }(\mathbf {x}) \frac{\partial ^2 \eta (\hat{\theta }_{TW}, \mathbf {x})}{\partial \theta _u \partial \theta _v} \mathrm {d}\mathbf {x}& = & \int _{\mathcal {X}} \mu (\mathbf {x}) \frac{\partial ^2 \eta (\hat{\theta }_{TW}, \mathbf {x})}{\partial \theta _u \partial \theta _v} \mathrm {d}\mathbf {x}+ o_p(n^{-1/2}),\\& \overset{p}{\rightarrow } & \int _{\mathcal {X}} \mu (\mathbf {x}) \frac{\partial ^2 \eta (\theta _{L^2}, \mathbf {x})}{\partial \theta _u \partial \theta _v} \mathrm {d}\mathbf {x},\) 
by the consistency of \(\hat{\theta }_{TW}\) . Hence
\(\frac{\partial ^2 \ell (\hat{\theta }_{TW}; \mathbf {y})}{\partial \theta \partial \theta ^{ \mathrm {\scriptscriptstyle T} }} \overset{p}{\rightarrow } \frac{\partial ^2 L_{L^2}(\theta _{L^2})}{\partial \theta \partial \theta ^{ \mathrm {\scriptscriptstyle T} }} = V,\) 
as required.

The sequence of functions \(\partial ^3 \ell _{TW}(\theta ; \mathbf {y})/\partial \theta _i \partial \theta _j \partial \theta _h\)  is uniformly bounded for \(i,j,h=1,\dots ,p\)  and \(\theta \in E\) .
This follows from condition C2.

